Theorem moabs 2571

Description: Absorption of existence condition by uniqueness. (Contributed by NM, 4-Nov-2002.) Shorten proof and avoid df-eu 2597. (Revised by BJ, 14-Oct-2022.)

Assertion

Ref	Expression
moabs	⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃*𝑥𝜑))

Proof of Theorem moabs

Step	Hyp	Ref	Expression
1		ax-1 6	. 2 ⊢ (∃*𝑥𝜑 → (∃𝑥𝜑 → ∃*𝑥𝜑))
2		nexmo 2569	. . 3 ⊢ (¬ ∃𝑥𝜑 → ∃*𝑥𝜑)
3		id 23	. . 3 ⊢ (∃*𝑥𝜑 → ∃*𝑥𝜑)
4	2, 3	ja 188	. 2 ⊢ ((∃𝑥𝜑 → ∃*𝑥𝜑) → ∃*𝑥𝜑)
5	1, 4	impbii 212	1 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃*𝑥𝜑))

Syntax hints:   → wi 4   ↔ wb 209  ∃wex 1809  ∃*wmo 2565

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1810  df-mo 2567

This theorem is referenced by:  mo3  2592  mo4  2594  moeu  2611  dffun7  6565  wl-mo3t  38209